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Better recitation formatting

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andy
2026-01-09 13:38:46 -05:00
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source/recitations/calculus-review.ptx
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@@ -11,6 +11,9 @@
All of these will make an appearance in the context of probability density functions, cumulative distribution functions, and probability calculations in continuous sample spaces, so it will be useful to refresh your memory and practice some calculations. All of these will make an appearance in the context of probability density functions, cumulative distribution functions, and probability calculations in continuous sample spaces, so it will be useful to refresh your memory and practice some calculations.
</p> </p>
</introduction> </introduction>
<page>
<!-- Exercises start here. --> <!-- Exercises start here. -->
<exercise> <exercise>
<statement> <statement>
@@ -22,6 +25,7 @@
</p> </p>
</statement> </statement>
</exercise> </exercise>
<exercise> <exercise>
<statement> <statement>
<p> <p>
@@ -32,54 +36,91 @@
</p> </p>
</statement> </statement>
</exercise> </exercise>
<exercise> <exercise>
<statement> <introduction>
<p> <p>
Practice with the following derivatives: Practice with the following derivatives:
<md>
<mrow> \frac{d}{dx}\left( 2x^2 + x^{-3} + 4\sqrt{x} \right) \amp = </mrow>
<mrow> \frac{d}{dx}\left( \frac{1}{x} + 3e^{2x} + x^e + \pi^e \right) \amp = </mrow>
</md>
</p> </p>
</statement> </introduction>
<task workspace="2cm">
<p>
<m>\displaystyle \frac{d}{dx}\left( 2x^2 + x^{-3} + 4\sqrt{x} \right) = </m>
</p>
</task>
<task workspace="2cm">
<p>
<m>\displaystyle \frac{d}{dx}\left( \frac{1}{x} + 3e^{2x} + x^e + \pi^e \right) = </m>
</p>
</task>
</exercise> </exercise>
<exercise> <exercise>
<statement> <introduction>
<p> <p>
Practice with the following antiderivatives: Practice with the following antiderivatives:
<md>
<mrow> \int 2x^2 + x^{-3} + 4\sqrt{x} \ dx \amp = </mrow>
<mrow> \int \frac{1}{x} + 3e^{2x} + x^e + \pi^e \ dx \amp = </mrow>
</md>
</p> </p>
</statement> </introduction>
</exercise>
<task workspace="2cm">
<p>
<m>\displaystyle \int 2x^2 + x^{-3} + 4\sqrt{x} \ dx = </m>
</p>
</task>
<task workspace="2cm">
<p>
<m>\displaystyle \int \frac{1}{x} + 3e^{2x} + x^e + \pi^e \ dx = </m>
</p>
</task>
</exercise>
</page>
<page>
<theorem xml:id="thm-FTC"> <theorem xml:id="thm-FTC">
<title>Fundamental Theorem of Calculus</title> <title>Fundamental Theorem of Calculus</title>
<statement> <statement>
<p> <p>
If <m>F(x)</m> is an antiderivative of <m>f(x)</m><mdash />i.e., if <m>F'(x) = f(x)</m><mdash />then: If <m>F(x)</m> is an antiderivative of <m>f(x)</m><mdash />i.e., if <m>F'(x) = f(x)</m><mdash />then:
<md> <md>
<mrow> \int_a^b f(x)\ dx = F(x)\bigg|_a^b = F(b) - F(a). </mrow> <mrow> \int_a^b f(x)\ dx = F(x)\bigg|_a^b = F(b) - F(a). </mrow>
</md> </md>
If <m>f(x)</m> is a probability density function (don't worry about what that means for now), then integrals such as <m>\int_a^b f(x)\ dx</m> are used to calculate certain probabilities, and so are function values of the form <m>F(x)</m>.
</p> </p>
</statement> </statement>
</theorem> </theorem>
<p>
If <m>f(x)</m> is a probability density function (don't worry about what that means for now), then integrals such as <m>\int_a^b f(x)\ dx</m> are used to calculate certain probabilities, as are values of the form <m>F(x)</m>.
</p>
<exercise> <exercise>
<statement> <introduction>
<p> <p>
Practice with the following integrals: Practice with the following integrals:
<md>
<mrow> \int_0^1 \frac{3}{2} x^{3/2} \ dx \amp = </mrow>
<mrow> \int_0^x 3e^{-3t} \ dt \amp = </mrow>
</md>
</p> </p>
</statement> </introduction>
</exercise>
<task workspace="2cm">
<p>
<m> \displaystyle \int_0^1 \frac{3}{2} x^{3/2} \ dx = </m>
</p>
</task>
<task workspace="2cm">
<p>
<m> \displaystyle \int_0^x 3e^{-3t} \ dt = </m>
</p>
</task>
</exercise>
</page>
</worksheet> </worksheet>